Linear Dynamic Equations Research Articles

Almost periodic solutions for neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scales

In this paper, a class of neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scales is considered. By utilizing the exponential dichotomy of linear dynamic equations on time scales, Banach’s fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of neural networks. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and the discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new and complementary to the previously known results even when the time scale .

Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales

In this paper, using the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, some sufficient conditions for the existence and global exponential stability of almost automorphic solutions for a class of neutral type high-order Hopfield neural networks (HHNNs) with delays in leakage terms on time scales are obtained. Finally, a numerical example is provided to illustrate the feasibility of our results and our results also show that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new even when the time scale T=R or Z and show that the delays in the leakage term do harm to the existence and global exponential stability of almost automorphic solutions.

Discrete-space partial dynamic equations on time scales and applications to stochastic processes

We consider a general class of discrete-space linear partial dynamic equations. The basic properties of solutions are provided (existence and uniqueness, sign preservation, maximum principle). Above all, we derive the following main results: first, we prove that the solutions depend continuously on the choice of the time scale. Second, we show that, under certain conditions, the solutions describe probability distributions of nonhomogeneous Markov processes, and that their time integrals remain the same for all underlying regular time scales.

Combining Methods of Lyapunov for Exponential Stability of Linear Dynamic Systems on Time Scales

Consider the linear dynamic equation on time scales (1) where , , is a rd-continuous function, T is a time scales. In this paper, we shall investigate some results for the exponential stability of the dynamic Equation (1) by combinating the first approximate method and the second method of Lyapunov.

Hyers-Ulam stability of abstract second order linear dynamic equations on time scales

In this paper we investigate the Hyers-Ulam Stability of the abstract dynamic equation of the form

Existence and Global Exponential Stability of Almost Periodic Solutions for a Class of Delay Duffing Equations on Time Scales

Based on the exponential dichotomy of linear dynamic equations on time scales, we obtain some sufficient conditions for the existence and global exponential stability of almost periodic solutions for a class of Duffing equations with time-varying delays on time scales. We also present numerical examples to show the feasibility of obtained results. The results of this paper are completely new even when the time scaleT=RorZand are complementary to the previously known results.

Periodic Solutions for Shunting Inhibitory Cellular Neural Networks of Neutral Type with Time-Varying Delays in the Leakage Term on Time Scales

A class of shunting inhibitory cellular neural networks of neutral type with time-varying delays in the leakage term on time scales is proposed. Based on the exponential dichotomy of linear dynamic equations on time scales, fixed point theorems, and calculus on time scales we obtain some sufficient conditions for the existence and global exponential stability of periodic solutions for that class of neural networks. The results of this paper are completely new and complementary to the previously known results even if the time scale𝕋=ℝorℤ. Moreover, we present illustrative numerical examples to show the feasibility of our results.

Simplified non-linear dynamical equations of circular cylindrical shells

Simplified non-linear dynamical equations of circular cylindrical shells are obtained on the basis of asymptotic analysis. The original partial differential equations (PDEs), which have eighth order in the longitudinal coordinate, are replaced by two sets of equations that have fourth order on this coordinate. The first set contains non-linear and dynamic PDEs. They describe the oscillations in the inner region of the shell. For small non-linearities, they coincide with the well-known linear semi-membrane dynamical equations. Ordinary differential equations of the edge effect allow one to satisfy all the boundary conditions. They are linear and quasistatic. An important question about the boundary conditions for the simplified equations is considered. We also discuss the solution of the obtained boundary value problems.

Volterra Theory on Time Scales

This paper is concerned with the existence and uniqueness of solutions to generalized Volterra integral equations on time scales. Unlike previous papers published on this subject, we can weaken the continuity property of the kernel function since the method we introduce here to guarantee existence and uniqueness does not make use of the Banach fixed point theorem. This allows us to construct a bridge between the solutions of Volterra integral equations and of dynamic equations. The paper also covers results concerning the following concepts: the notion of the resolvent kernel, and its role in the formulation of the solution, the reciprocity property of kernels, Picard iterates, the relation between linear dynamic equations and Volterra integral equations, and some special types of kernels together with several illustrative examples.

A class of linear dynamical equations for a Banach space on time scales

In this paper we consider the existence and uniqueness of global solutions to linear dynamical equations for a Banach space on time scales from a new point of view. We characterize those linear dynamical equations for a Banach space whose existence and uniqueness of global solutions do not depend on concrete time scales.MSC:34G10, 34K30, 39A13.

다물체 동역학 해석방법을 이용한 철도차량의 임계속도 계산

본 연구에서는 다물체 동역학 모델을 이용한 철도차량의 임계속도 계산 방법을 제시하였다. 휠과 레일의 접촉 구속조건과 접촉력을 휠셋 단위에서 수식화하였다. 이를 대차모델에 합하여 구속조건을 가진 다물체 동역학 운동방정식으로 철도차량의 동적모델을 표현하였다. 철도차량의 다물체 동역학 모델에 대한 비선형 구속조건식과 운동방정식은 QR 분해법을 이용하여 독립좌표만으로 이루어진 선형방정식으로 유도하였다. 유도된 선형방정식으로부터 휠셋 및 이륜 대차에 대한 고유치 해석결과를 통해 임계속도를 구하였다. 임계속도에 영향을 미치는 차량 파라미터의 영향에 대한 결과를 제시하였다.

Stability Analysis of High-Speed Railway Vehicle Based on Multibody Dynamics Analysis

Multibody dynamics analysis is advantageous in that it uses real dimensions and design parameters. In this study, the stability analysis of a railway vehicle based on multibody dynamics analysis is presented. The equations for the contact points and contact forces between the wheel and the rail are derived using a wheelset model. The dynamics equations of the wheelset are combined with the dynamics equations of the other parts of the railway vehicle, which are obtained by general multibody dynamics analysis. The equations of motion of the railway vehicle are linearized by using the perturbation method. The eigenvalues of these linear dynamics equations are calculated and the critical speed is found.

On initial and boundary value problems for implicit dynamic equations on time scales

This paper is concerned with the index concept, the unique solvability of the initial value problem, the two point boundary-value problem and Green’s function for a class of linear implicit dynamic equations with index-1 on time scales. The results are a generalization of the previous ones for differential and difference-algebraic equations.

Some qualitative properties of linear dynamic equations with multiple delays

This paper discusses stability and asymptotic properties of the delay dynamic equation y ∇ (t)= ∑ j = 0 k a j y ( τ j ( t ) ) ,t∈T, where a j ∈R are scalars, τ j are iterates of a function τ:T→T and T is a time scale unbounded above. Under some specific choices of τ, this dynamic equation involves several significant particular cases such as linear autonomous differential equations with several delays or linear autonomous higher-order difference equations. For proportional τ, we formulate an asymptotic result joint for two different time scales, including a joint form of its proof. For a general τ, we investigate stability and asymptotic properties of solutions on the continuous and discrete time scales separately. Besides a related character of the relevant results, we discuss also a possible related character of their proofs.MSC:34N05, 34K25, 39A12, 39A30.

The method of finding solutions of partial dynamic equations on time scales

On time scales, one area lacking of development is the method of finding solutions on partial dynamic equations. This paper proposes a method for finding the exact solution of linear partial dynamic equations on arbitrage time scales. We modify the variational iteration method on ℝ to find an approximation of the nonlinear partial dynamic equation on q N ¯ . As an example, the modified variational iteration method is applied to q-Berger equations and to q-Fisher equations. Their numerical results reveal that the proposed method is very effective.

DGFEM for dynamical systems describing interaction of compressible fluid and structures

The paper is concerned with the numerical solution of flow-induced vibrations of elastic structures. The dependence on time of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian–Eulerian) formulation of the compressible Navier–Stokes equations. The deformation of the elastic body, caused by aeroelastic forces, is described by the linear dynamical elasticity equations. These two systems are coupled by transmission conditions. The flow problem is discretized by the discontinuous Galerkin finite element method (DGFEM) in space and by the backward difference formula (BDF) in time. The structural problem is discretized by conforming finite elements and the Newmark method. The fluid–structure interaction is realized via weak or strong coupling algorithms. The developed technique is tested by numerical experiments and applied to the simulation of vibrations of vocal folds during phonation onset.

LINEAR TRANSFORMATIONS AND FLOQUET THEOREM FOR LINEAR IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES

This paper is concerned with Cauchy problem, Lyapunov transformations, Floquet and Lyapunov theorems for linear implicit dynamic equation AtxΔ= Btx with index-1 on time scales. The stability of this equation under the act of these Lyapunov transformations is also considered. The results are a unification and generalization of previous results for differential-algebraic equations and implicit difference systems.

Variable Structure Control for Moving Mass Actuated Reentry Vehicles

Dynamic equations of an axisymmetric moving mass actuated reentry vehicle, which was in the state of tri-channel stability, were derived in this paper, and the linear dynamic equations in longitudinal plane were obtained. Then variable structure control method was applied to control the moving mass actuated reentry vehicle. The simulation results indicate that the controller is of satisfactory performances in the nonlinear system.

Rational decay rates for a PDE heat--structure interaction: A frequency domain approach

In this paper, we consider a simplified version of a fluid--structure PDEmodel---in fact, a heat--structure interaction PDE-model. It is intended to be a first step toward a more realistic fluid--structure PDE model which has been of longstanding interest within the mathematical and biological sciences [33, p. 121], [17], [19]. This physically more sound and mathematically more challenging model will be treated in [13]. The simplified model replaces the linear dynamic Stokes equation with a linear $n$-dimensional heat equation (heat--structure interaction). The entire dynamics manifests both hyperbolic and parabolic features. Our main result is as follows: Given smooth initial data---i.e., data in the domain of the associated semigroup generator---the corresponding solutions decay at the rate $o( t^{-\frac{1}{2}}) $(see Theorem 1.3 below). The basis of our proof is the recently derived resolvent criterion in [15]. In order to apply it, however, suitable PDE-estimates need to be established for each component by also making critical use of the interface conditions. A companion paper [6] will sharpen Lemma 5.8 of the present work by use of a lengthy and technical microlocal argument as in [26,29,30,31], to obtain the optimal value $\alpha =1$; hence, the optimal decay rate $o(t^{-1})$. See Remarks 1.2,1.3.

Almost Periodic Solutions for Neutral-Type BAM Neural Networks with Delays on Time Scales

Using the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, we obtain some sufficient conditions for the existence and exponential stability of almost periodic solutions for a class of neutral-type BAM neural networks with delays on time scales. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new even if the time scale𝕋=ℝorℤand complementary to the previously known results.